48 research outputs found

    An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization

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    In this paper we develop an axiomatic setup for algorithmic homological algebra of Abelian categories. This is done by exhibiting all existential quantifiers entering the definition of an Abelian category, which for the sake of computability need to be turned into constructive ones. We do this explicitly for the often-studied example Abelian category of finitely presented modules over a so-called computable ring RR, i.e., a ring with an explicit algorithm to solve one-sided (in)homogeneous linear systems over RR. For a finitely generated maximal ideal m\mathfrak{m} in a commutative ring RR we show how solving (in)homogeneous linear systems over RmR_{\mathfrak{m}} can be reduced to solving associated systems over RR. Hence, the computability of RR implies that of RmR_{\mathfrak{m}}. As a corollary we obtain the computability of the category of finitely presented RmR_{\mathfrak{m}}-modules as an Abelian category, without the need of a Mora-like algorithm. The reduction also yields, as a by-product, a complexity estimation for the ideal membership problem over local polynomial rings. Finally, in the case of localized polynomial rings we demonstrate the computational advantage of our homologically motivated alternative approach in comparison to an existing implementation of Mora's algorithm.Comment: Fixed a typo in the proof of Lemma 4.3 spotted by Sebastian Posu

    On the computation of π\pi-flat outputs for differential-delay systems

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    We introduce a new definition of π\pi-flatness for linear differential delay systems with time-varying coefficients. We characterize π\pi- and π\pi-0-flat outputs and provide an algorithm to efficiently compute such outputs. We present an academic example of motion planning to discuss the pertinence of the approach.Comment: Minor corrections to fit with the journal versio

    Toric Ideals, Polytopes, and Convex Neural Codes

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    How does the brain encode the spatial structure of the external world? A partial answer comes through place cells, hippocampal neurons which become associated to approximately convex regions of the world known as their place fields. When an organism is in the place field of some place cell, that cell will fire at an increased rate. A neural code describes the set of firing patterns observed in a set of neurons in terms of which subsets fire together and which do not. If the neurons the code describes are place cells, then the neural code gives some information about the relationships between the place fields–for instance, two place fields intersect if and only if their associated place cells fire together. Since place fields are convex, we are interested in determining which neural codes can be realized with convex sets and in finding convex sets which generate a given neural code when taken as place fields. To this end, we study algebraic invariants associated to neural codes, such as neural ideals and toric ideals. We work with a special class of convex codes, known as inductively pierced codes, and seek to identify these codes through the Gröbner bases of their toric ideals

    Une approche par l’analyse algĂ©brique effectivedes systĂšmes linĂ©aires sur des algĂšbres de Ore

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    The purpose of this paper is to present a survey on the effective algebraic analysis approach to linear systems theory with applications to control theory and mathematical physics. In particular, we show how the combination of effective methods of computer algebra - based on Gröbner basis techniques over a class of noncommutative polynomial rings of functional operators called Ore algebras - and constructive aspects of module theory and homological algebra enables the characterization of structural properties of linear functional systems. Algorithms are given and a dedicated implementation, called OreAlgebraicAnalysis, based on the Mathematica package HolonomicFunctions, is demonstrated.Le but de ce papier est de prĂ©senter un Ă©tat de l’art d’une approche par l’analyse algĂ©brique effective de la thĂ©orie des systĂšmes linĂ©aires avec des applications Ă  la thĂ©orie du contrĂŽle et Ă  la physique mathĂ©matique.En particulier, nous montrons comment la combinaison des mĂ©thodes effectives de calcul formel - basĂ©es sur lestechniques de bases de Gröbner sur une classe d’algĂšbres polynomiales noncommutatives d’opĂ©rateurs fonctionnels appelĂ©e algĂšbres de Ore - et d’aspects constructifs de thĂ©orie des modules et d’algĂšbre homologique permet lacaractĂ©risation de propriĂ©tĂ©s structurelles des systĂšmes linĂ©aires fonctionnels. Des algorithmes sont donnĂ©s et uneimplĂ©mentation dĂ©diĂ©e, appelĂ©e OREALGEBRAICANALYSIS, basĂ©e sur le package Mathematica HOLONOMIC-FUNCTIONS, est prĂ©sentĂ©

    On the Rapoport-Zink space for GU(2,4)\mathrm{GU}(2, 4) over a ramified prime

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    In this work, we study the supersingular locus of the Shimura variety associated to the unitary group GU(2,4)\mathrm{GU}(2,4) over a ramified prime. We show that the associated Rapoport-Zink space is flat, and we give an explicit description of the irreducible components of the reduction modulo pp of the basic locus. In particular, we show that these are universally homeomorphic to either a generalized Deligne-Lusztig variety for a symplectic group or to the closure of a vector bundle over a classical Deligne-Lusztig variety for an orthogonal group. Our results are confirmed in the group-theoretical setting by the reduction method \`a la Deligne and Lusztig and the study of the admissible set

    Key problems in the extension of module-behaviour duality

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    AbstractThe duality for linear constant coefficient partial differential equations between behaviours and finitely generated modules over the operator ring is a very powerful tool linking equation structure to dynamic behaviour. This duality is critically dependent on the choice of signal space. In this paper we discuss two key algebraic problems which form an obstacle to the extension of this theory to general signal spaces. The first of these is the so-called Willems closure problem, which limits the ability of system equations to directly describe the system. The second is the elimination problem, the general solution of which depends upon an algebraic property (injectivity) of the signal space. We demonstrate the importance of these problems in the module-behaviour framework, and some of the useful consequences of a full or partial solution. The issues here are of particular relevance to the extension of the current duality theory for behaviours defined by linear partial differential equations from the case of constant to non-constant coefficients
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